DR. BOOTH ON THE GEOMETRICAL PROPERTIES OF ELLIPTIC INTEGRALS. 369 
Section VI. 
XLIX. The difference between an arc of a logarithmic hyperbola, and the corre- 
sponding arc of the tangent paj'abola, may he expressed by the arcs of a plane, a spherical 
and a logarithmic ellipse. 
. , r du T rcos®(pd<p 
Resuming- the equation (241.), ✓b(A+C)J 
and combining (248.) with (249.), we may easily show that 
n(l — m) mn^ 
'v/B(A + C) m-\-n — 2nm ’ 
and from (258.) we deduce that 
(281.) 
G 
Let 
M vr 
= (282.) 
Substituting this value ofj'- 
vious reductions, 
’cos^ipdip . 
2f-^ 
Jcos'^‘0 
M2 VI 
2T 
in the preceding equation we get, after some oh- 
mn p, ”(1 
k m + n — 2mn 
V mn . j M V 1 
Now a, and h, being the semiaxes of the base of an elliptic cylinder whose curve of 
section with the paraboloid is a logarithmic ellipse, let, as in (171-)? 
mn{\—m) }p mn{\—n) 
p («— m)2 ’ ~j^ (n— m)2 ’ 
(283.) 
and if we put 2 for an arc of this logarithmic ellipse, we shall have, as in (163.), 
22 V mrip, 
n{l—m) 
C df 
^ n — m 
V mn ^ 
]m VI 
Subtracting this equation from the preceding, and replacing G by its value in (282.), 
we finally obtain 
(284.) 
Jcos'^o jcos'^T {n—m)[m + n—2nm) 
We may express the arc T by the help of one parabolic arc only, if we introduce the 
r dr 
equation established in (160.), S=2+^) — r, hence 
• (2S5.) 
1 TT 
When sin<p=^^, o=-^, and the arc of the logarithmic hyperbola becomes infinite, the 
arc of the parabola also becomes infinite, and an asymptot to the logarithmic hyper- 
3 B 2 
